Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $p = \dfrac{27t - 18}{5t} \div \dfrac{12t - 8}{2t} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{27t - 18}{5t} \times \dfrac{2t}{12t - 8} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ (27t - 18) \times 2t } { 5t \times (12t - 8) } $ $ p = \dfrac {2t \times 9(3t - 2)} {5t \times 4(3t - 2)} $ $ p = \dfrac{18t(3t - 2)}{20t(3t - 2)} $ We can cancel the $3t - 2$ so long as $3t - 2 \neq 0$ Therefore $t \neq \dfrac{2}{3}$ $p = \dfrac{18t \cancel{(3t - 2})}{20t \cancel{(3t - 2)}} = \dfrac{18t}{20t} = \dfrac{9}{10} $